Artificial network



May 15, 1-934. w A 1,958,742

ARTIFICIAL NETWORK Filed Dec. 1, 1930 3-Sheets-Sheet 2 /N VE'NTO.R WILHE'LM Caz/E A TTORNEY Patented May 15, 1934 PATENT OFFICE ARTIFICIAL NETWORK Wilhelm Cauer, Gottingen, Germany Application December 1, 1930, Serial No. 499,233 In Germany June 8, 1928 21 Claims.

tates the impedance of a given long line or a combination of lines is an' artificial network because it is designed to approximate to the frequency characteristics of the long line or the combination of lines. An attenuation-correction circuit or a phase-correction circuit is an artificial network because it produces the desired attenuation or phase correction as a function of frequency.

In the case of a filter, let it be assumed, for

example, that it is desired to have zero attenua tion in the transmission ranges and infinite attenuation in the attenuating ranges. It is impossible. usually, to attain the ideal conditions exactly, but a filter that will attain or approximate to the desired frequency characteristics is an artificial network.

An object of the invention is to provide a new and improved method of designing circuits equivalent to other circuits.

A further object is to improve upon lines and circuits of the above-described character with the end in view of improving their efficiency and field of utility and reducing their cost of manufacture.

Another object is to provide a new and improved coupling of inductances in the said circuits and, in particular, tight coupling.

Other and further objects will be explained hereinafter and pointed out in the appended claims, it being understood that I intend to claim all the novelty that my invention may possess.

The invention will now be explained in connection with the accompanying drawings, in which Fig. 1 is a diagrammatic view of a canonical form of a general, four-terminal network without resistances; Fig. 2 is a similar view of a network that is reciprocal or inverse to the network of Fig. 1, the coils being mutuallyinductively related; Fig. 3 is a similar view of a network all the mutual inductances of which are tightly coupled; Fig. 4 is a diagram of a canonical form of a four-terminal network having inductance, capacity and resistance; Fig. 5 is a diagrammatic view of a canonical form of a twoterminal network, all the coils being mutually inductively related; Fig. 6 is a view of a network reciprocal or inverse to Fig. 5;'Fig. '7 is a View of a two-terminal network having impedances with mutual inductances; Fig. 8 is a diagrammatic view of a wave-filter circuit; Fig. 9 is a corresponding view of a network which constitutes an improvement over the network .of Fig. 8; Fig. 10 is a diagram for aiding the understanding of the invention; Fig. 11 is a diagrammatic view of a reactive four-terminal network; Figs. 12 and 13 are similar views of two-terminal portions that may be embodied in Fig. 11-; Fig. 14 is a view of a network equivalent to Fig. 11 and having mutual inductance; Figs. 15 and 16 are views of two-terminal portions that may be embodied in Fig. 11 alternatively to the portions illustrated in Figs. 12 and 13, respectively; Figs. 17 and 18 are views of networks, with mutual inductance, eachequivalent to the network of Fig. 11 when the latter embodiesthe two-terminal portions illustrated in Figs. 15 and 16, Fig. 1'7 being a special case of Fig. 14; and Figs. 19 and 20 are views of two equivalent phase-compensating networks, Fig. 20 affording also a numerical example illustrative of the invention.

Some of 'the terminology hereinafter employed will first be explained in connection with Fig. 10, showing a general, two-terminal network containing two meshes, the terminals being indicated at 1, 1. The invention, of course, is general, and is not restricted either to two-terminal networks or to networks having two meshes, but this simple case will serve for illustrative purposes.

The first mesh of the network of Fig. 10 is shown provided with series-arranged inductance 3, resistance 4, and capacitance 5. The inductance 3 has a value'L1, the resistance 4 has a value R1 and the capacitance 5 has an inverse or reciprocal value D1.

In the second mesh, similarly, there is inductance 6, resistance 7 and capacitance-8, denoted by L3, R3 and D3, which symbols represent also the values of the inductance and the resistance and the inverse or reciprocal value of the capacitance in the said second mesh.

A branch between the two meshes is illustrated as having inductance 9, resistance 10 and capa i tance 11, and having, respectively, the values L2, R2 and the reciprocal of D2.

The value of the .mutual inductance between the inductance L1 and L2 will be represented by M12; that between the inductance L2 and La by M23; and that between the inductance L1 aud L: by M13.

The current flowing in the first mesh will be denoted by I1 and that in the second mesh by I2.

The voltage across the two terminals 1, 1 will be referred to as E1. The driving-point impedance 2, then, will be It will be useful to use also the following symbols, which relate to meshes as the L1, L2, relate to branches:

Similar symbols will be used for more compli cated artificial networks having a larger number of meshes. The complex currents flowing in the successive meshes E, II, 1H, n

will be denoted by 11, I2, I3, n

respectively. The inductances and the mutual inductances between the meshes s and t will be denoted by La. Thus, the symbol for, the inductances and the mutual inductances between the first and second meshes will be represented by L12. The corresponding resistance will be denoted by Rst and the corresponding reciprocal or inverse capacity by Dst. The pulsatance will be represented, as usuaL'by 10, where w=27rf I, being the frequency of the electromotive force in the first mesh. The following customary abbreviation will also be used:

' A=jw where 1' -1. A very brief outline of the theory of equivalence will'now be given. For a fuller discussion, reference may be had to my paper, entitled Eine Klasse von Funktionen, die die Sheltjesschen Kettenbruche als Sonderfall enthalt, in Jahres bericht der deutschen Mathematikervereinigung,

A general-passive network of the character heretofore described, with n independent meshes,

is determined, in the steady state, by ing equations:

A11 I1+A12 Iz-l-Ara 13+ A21 I1+A22 I2+A23 13+ the follow- 11 12 ia ln 21 22 23 2 A-i A A,. ---A.-

and

Then the driving-point impedance for the first mesh is s t e t,

s, t l where :08 and .L't are any variables, by the affine, real transformation:

$1=1' E2=a21w1+a22$2'+a23$3'+ +a2n$n x3=a31a21'+a32$2+u33333'+ +d3n$n' where :25 is likewise any variable, and where a is an arbitrary, real constant. Then and I n I n I I I at a t A si a s, t2 s, =2 The new coefficients A'st, where at a al+ st thus represent a new networkjhavlng new inductances L'st, new resistances R'St and new reciprocal capacities D'st. The new coefficients may be calculated by known algebraic methods such, for example, as the multiplication of matrices.

I nn network is therefore one equivalent of the original network.

An important fact to be noticed is that, in the new network, as in the original network, the quadratic form representing the energy of the capacities, satisfy the necessary conditions that they be positive definite.

It follows that the above group of linear transformations fundamentally determines the theory of equivalence of driving-point impedances. The

theory includes, for example, as a special case,

the discussion of the driving-point impedances of two-mesh circuits considered by Foster in the Bell System Technical Journal, 1924, and by me in Archiv fur Elektrotechnik, 1926. As will be clear by reference to my paper entitled, Untersuchungen iiber ein Problem, das drei positiv definite quadratische Formen mit Streckenkomplexen in Beziehung setzt, in Mathematische Annalen, Band 105, Heft 1, 1931, indeed, the complete theory is somewhat more complicated. This is particularly because the above-given necessary conditions for the resistances and capacities are not suificient, for there is no such thing possible physically as a mutual resistance and a mutual capacity, with the result that the transformed network is not always immediately capable of physical realization in the precise form determined by the linear transformation. This exception, however, is more apparent than real, because it is possible, as is also explained in my said paper, to produce an actual, physical network in all cases,even the cases mentioned of mutual resistances and capacities,by suitable further modification and interpretation. With this explanation, therefore, it will be understood that the theory is perfectly general. The very fact that there are no physical mutual resistances and capacities, on the other hand, emphasizes the importance of those cases involving the use of mutual inductances. There are other complications also, particularly in the case where all three kinds of impedances, namely, inductance, capacitance and resistance, are found in the network.

It will be obvious that to every network there correspond a large number of equivalent networks. Some of these have special properties that make them more advantageous for use in particular cases than others. For example, if some of the coeflicients are zero, the corresponding network will have a fewer number of elements,,such as ind'uctances, capacitances, or resistances. In particular, it is possible to choose a network equivalent to some other network and that shall have the minimum number of elements. This result may be obtained by the same lineartransformation method, as before described, and is similar to the well-known method for reducing two quadratic forms, by means of a general linear transformation, to a sum of squares. As an illustration, let the following linear transformation be applied to a four-terminal network in which I1 is the current at the input terminals 1, 1 and I2 at the output terminals 2, 2. This network is to be perfectly general except that it is to have no resistances:

332::62' $3=a31$1+a32$2+a33333'+ |a3n1vn' 9o m4=a41$1'+u42.732'+a431'3+ +a4nwn' I1;=a1r1$1+un2922'+an3:123'+ +(lnnl'n Any given pair of matrices (La), (Dst) may be reduced, by the same transformation, to the forms L11 L12 L33 L44 Lnn L21 L22 L23 L24 L211 L33L32L330 ...0 L44 L42 L44 0 inn n L1i2 0 0 Lnn and D11 D120 ...0 D21 D22 0 0 0 0 D3 0 respectively. Then the canonical form of circuit illustrated in Fig. 1 will be obtained (if greater details are desired, they will be found in my said paper in Mathematische Annalen, particularly commencing with page 121). This circuit comprises a number of parallel-connected coils and condensers 12, 13 arranged in series with a coil 14 and a condenser 15, the last-named condenser 15 having a shunt-arranged coil 16 and condenser 17. An ideal transformer T is connected in this shunt circuit. By ideal is meant a tightlycoupled transformer having great inductance. The arrows connect coils that are mutually inductively related, and only the coils so indicated 125 by the arrows are so related.

The numerical values of the transformed ele ments Lst, D5: of the new network obtained from the original network, may be calculated also by other methods, and even without knowledge of the reducing transformation. The following is one method that may be employed (and it will be understood that the same method is applicable also to the treatment of certain other canonical forms) Let V (112 (1.21

represent the minor of the determinant a that is complementary to the element A12; let azz represent the corresponding minorcomplementary to the element A22; and let aim represent the minor which is the algebraic complement of A11 A12 A21 A22 1122 that is, the impedance obtained across the ter- 5 minals 1, 1 in Fig. 1, for example, when the terminals 2, 2 are open. In the circuit of this Fig. 1,this may be effected by resolution into partial fractions. In order not to obtain physi-=- cally unrealizable mutual capacities, I shall assume that the mesh 2 is-not capacitatively connected with the meshes III, IV n. As'for the quantities.

D21, L21, L23, L24 Lzn they may be determined uniquely by means of a system of linear equations, derived by comparison of the coefiicients of the various powers of A in A am. Finally, L22 and D22 may be uniquely calculated from an. The network constants so obtained will always correspond to physically realizable circuits.

The only case that might at all be considered troublesome is when all networks, whether two-terminal, four-ter-.

minal, etc.

First, in many cases, the economy of the network system is greatly improved if the network contains at least three mutually inductively related coils, no two of which are directly galvanically connected together. For example, in Fig. 3, the 00181013, 1023, was, as will be explained hereinafter, are no two of them galvanically connected together, and they are mutually inductively related.

Secondly, it is possible to reduce any network with any frequency characteristics to equivalent canonical forms.

Thirdly, among these canonical forms, for any network, it is always possible to find some that have all their mutual inductances tightly coupled.

Fourthly, corresponding to every two-terminal or four-terminal network (and also other networks with corresponding definitions) having mutual inductance, it is always possible to design a reciprocal or inverse network. The possibility of constructing reciprocal four-terminal networks may be useful also in such cases where two reciprocal networks are not found in a larger network. For a four-terminal network, the definitionof a reciprocal or inverse network is as follows:

Let gm be the impedance across the terminals 1. 1 when the terminals 2, 2 are short circuited, and 311 0 be the impedance when the terminals 2,2 are open circuited. Secondly, let ya: and yzobe the short-circuit and the open-circuit impedances at the terminals 2, 2. Let 21k, 210 and 22:, 220 be the-corresponding impedances for the second network. Then the second network .is reciprocal or inverse to the first if any other number of terminals. Thus, in the like each branch 12 case of the reciprocal or the inverse of a twoterminal network, the product of the impedances which are known as reciprocal to each other is a positive constant, the square of a resistance. The general definition of the reciprocal of a Zn-terminal network may be found in my paper in the Mathematische Annalen, Vol. 105 (1931) and in my paper New Theory and Design of Wave-Filters, Physics 1932, especially Appendix II, at the top of page 265. As indicated on this page 2 55, the reciprocal of a 2n-terminal network is a network whose characteristic matrix is the reciprocal of the characteristic matrix of the original network, except for a positive, constant factor.

It is, of course, a physical impossibility to enumerate and describe herein all the numerous equivalent two-terminal or four-terminal network circuits. Only a few examples can here be given.

Fig. 1, for example, as already stated, represents a canonical, four-terminal network having other four-terminal network without resistances 4 with n independent meshes. It may have many uses, such as for wave-filter purposes. As a filter, it is, in general, equivalent to any unsymmetrical wave filter having n independent meshes, the circuit elements being suitably chosen and including mutual inductances. The 311 elements of the circuit all haveindependent influence on the frequency characteristics. The transformer T should have, very. great inductance and negligible stray, as before described.

The four-terminal network of Fig. 2 is reciprocal or inverse to that of Fig. 1. The coils 18, 20 and 23 are connected by mutual inductances. The transformer T is of the same character as that of Fig. 1. The reciprocal or inverse circuit of Fig. 2 is obtained from that of Fig. 1 in the the following manner. The canonical circuit of Fig. 1 is first replaced by an imaginary equivalent network in which no coils by interchanging the role of the capacities and the inductances. That is, in the imaginary networkjthe coils 12, 14,16, which are mutually related, but no two of which are directly connected together galvanically, are replaced by interlinked capacities, and the capacities 13, 15, 17 are replaced by non-interlinked inductances. The imaginary .circuit thus produced is then replaced, in a known'manner, by another arrangement, branch for branch,branches in parallel, and the corresponding branch 13 of Fig. 1, being replaced by branches in series, like each branch 19 and the corre-.

spending branch 18 of Fig. 2, and vice versa. The inductances thus become replaced by proportional capacities. The quadratic formorm the coefllcients thereof represent therefore inductances and mutual inductances that are always physically realizable. In the reciprocal or inverse circuit of Fig. 2, the inductance 23 corresponds to the imaginary capacity corresponding to the inductance 14 of Fig. 1. The capacity 21 of Fig. 2,'similarly, corresponds to the capacity 17 of Fig. 1; the capacity 22 of Fig. 2 to the capacity 15 of Fig. 1; and the inductance 20 of Fig. 2 to the inductance 16 of Fig. 1.

In the circuit arrangement of Fig. 3, there are 11. series of coils each series being connected between two terminals 1, 1; 2, 2; 3, 3 n, 11., respectively. The first series of coils is indicated at $011, 1012, 1013 win;

the second series at the third series at was, w34 w3n,

etc.

The last series consists of a single coil wnn. The coils of the second series are tightly connected with each coil of the first series except one; the coils of the third series are tightly connected with each coil of the second series except one, and with all those of the first series except two; and so on. The last series, which consists of only one coil 101111, is tightly connected with one coil of each of the other series.

The network is, in general, equivalent to n coils arbitrarily interlinked, with inductances Lst, and is therefore a canonical form for n arbitrarily interlinked coils. The couplings are all ideal and tight, with 7.051: turns. The inductances of the primary coils are For n=3, the following relations hold goodbetween L, 10st and Lstl A more complicated case, including resistances as well as inductances and capacitances, is illustrated by the canonical, four-terminal network of Fig. 4 and is equivalent to every four-terminal network constituted of three independent meshes, The diagram is restricted to only three independent meshes, for the sake of simplicity only,

- in order not to complicate the drawing, the same 34, 35, 36. The coils 25,32 form :a tightly coupled transformer S1. The coils 26, 33 and a coil 37 form a tightly coupled transformer S2. Coils 38, 39 and 40 are connected in shunt across the resistances 28 and 29 and the condenser 30, respectively, in the meshes A, B and C. If the meshes A, B and C are omitted, the circuit will have exactly three independent meshes. The coils 34 and 38 constitute a tightly coupled transformer T1. The coils-35 and 39 and a coil 41 constitute a tightly coupled transformer T2. The coils 36 and 40 constitute a tightly coupled transformer T3. The transformers T1, T2 and T3 are ideal transformers.

The coils 37 and 41 are connected in a circuit III in series with a condenser 42. This forms a canonical circuit with tight coupling. It is an advantage to have a circuit with tight coupling because it is more easily realizable in practice with given values of the elements. The necessity of taking the transformers T1, T2 and T3 arises from the fact that no mutual capacities or resistances exist.. The importance lies, not so much in having the transformers T1, T2 and Ts ideal, as in having mutual inductances. This is true in general, and is not restricted to the particular circuit shown.

In the most general case, three independent meshes will involve 18 elements. If we count several coils on a single core, as one coil, and if we use the circuit arrangement of Fig. 4, then the 18 elements will reduce to 12. These 12 are as follows:

(1) Coil 24;

(2) The transformer S1;

(3) The transformer S2;

(4), (5), (6) The resistances 27, 28, 29; (7), (8), (9) The transformers T1, T2, T3; (10) The condenser 42; (11) The condenser 30; and

(12) The condenser 31.

It will be noted that, in the transformer S2, as in the case of the coils U713, 11223 and was of Fig. 3, there are three coils, not immediately galvanically connected together, but all tightly coupled together. The same is true of transformer T2.

In Fig. 5, there is shown a canonical two-terminal network, the coils 43 having inductance, and the network having also mutual inductance, capacities and resistances. By our definition, this circuit is equivalent to all other two-terminal circuits with n independent meshes containing inductances, including mutual inductance, ca.-

pacities and resistances. Each mesh has a coil 43 and a condenser 44 in series and a resistance 45 in shunt. The coils 43 are all interlinked by mutual inductances.

The network of Fig. 6 is reciprocal or inverse to that of Fig. 5, the analogous relations being similar, but not exact, to that existing between Figs. 1 and 2, and the mutual inductances of the reciprocal or inverse'circuit being calculated in the same manner as before described.

To obtain the imaginary circuit equivalent to the circuit of Fig. 5, I replace the capacities 44 by resistances, the inductances 43 by imaginary capacities which are interlinked, and the resistances 45 by non-interlinked inductances.

The said imaginary circuit may be obtained by using the principle of linear transformation'for the given circuit. The coils 47*are mutually interlinked, the Las being determined by the quadratic form which is reciprocal or inverse to the quadratic form of the DstS of the imaginary circuit equivalent to the circuit of Fig. 5. This imaginary circuit will correspond to the reciprocal or inverse circuit. Fig. 6, in the same manner as described above in connection with Figs. 1 and 2. This circuit of Fig. 6 contains condenser 46 in series and inductances 47 and resistances 48 in parallel.

In Fig. 7, one of the networks Z1 has two terminals 49 and 50 and the other has two terminals 51 and 52. The networks Z2 have each the two terminals 49 and 52, and 50*and 51, respectively. The two-terminal networks Z1 and Z2 contain mutual inductance, satisfying the relation Where R is a constant, equal to the value of the resistance across the terminals 50 and 52. The practical value of this arrangement is that, given any impedance Z1, it is possible to design a second impedance Z2 so as to have a constant resistance similarly each have a condenser 55, an inductance c and an inductance d in parallel with a condenser 56. The coils a, a are mounted on a single core. The same is true of the coils b, b, aswell as of the coils c, c and the coils d, d. This lattice-type network is useful as a wavefilter circuit if the elements a, b, c, d, 53, 54, 55 and 56 are suitably chosen.

Fig. 9 illustrates a balanced four-terminal network circuit comprising a multiple-inductance wave filter embodying the present invention. A balanced network, as is well understood in the art, involves symmetry along the horizontal line. The network of Fig. 9 can be made equivalent to the filter of Fig. 8 by suitable choice of dimensions of the elements and which contains a smaller number of capacities and also a more suitable choice of dimensions of the capacities than is true of the circuit of Fig. 8. This circuit comprises four branches, as in Fig. 8. The first branch contains a condenser 57, and two coils 58, 59 in series between the terminals 1, 2. The second branch, between the terminals 1, 2, contains a condenser 60 and two coils 61 and 62 in series. The third branch contains a condenser 63 and two coils 64, 65 between the terminals 1', 2'. The fourth branch contains a condenser 66 and two coils 67 and 68 between the terminals 1', 2.

The coils 58 and 64 are tightly coupled and mounted upon a single core 69. The same is true of the coils 61 and 67, upon a single core '70. The coils 59 and 65 are similarly mounted. upon a single core 71 with a third coil '72, which latter is shunted by'a condenser '73. It will be noted that the coils 59, 65 and 72 are none of them directly galvanically connected, but are mutually tightly coupled. A similar construction obtains for the coils 62 and 68 and a third coil '74, which latter is shunted by a condenser 75. The coils 62, 68 and '74 are tightly coupled and are all mounted upon a single core 76 similarly to the coils 59, 65 and 72.

The six condensers 57, 60, 63, 66, 73 and 75 are smaller in number than the eight condensers 53, 54, 55 and 566i Fig. 8. A further advantage of the circuit of Fig. 8 is to avoid too great capacities such as would be necessary, in'practice, with the circuit of Fig. 8. The dimensions of the elements indicated in Fig. 9 were calculated Dem?- hand for a given attenuation requirement and were actually checked experimentally.

The networks disclosed, for example, in Figs. 9

and 20, being symmetrical about a median, horizontal line, are balanced with respect to the ground potential. Thus, in Fig. 20, the middle point of the right-hand coil 5 may be connected with the ground. To avoid confusion, it may be stated that the expression symmetrical fourterminal network, as used in this specification, is not synonymous with balanced network, as

it assumes that the efiiciency of the four-terminal network is unafiected by an interchange of the input and output terminals. A network may, of course, be both balanced and symmetrical, as is the case with the networks illustrated in Figs. 9 and 20.

It would be a practical impossibility to illustrate all the many uses of my invention. Filters have already been mentioned, and a suggestion has also been made in connection with Fig. '7. The invention finds use also in simulating or corregtion or compensating circuits, as well as in other relations. According to present practice, particularly in the more complicated cases, the known methods of correction are not always successful. It is possible to attain success by using mutual inductances in accordance with my invention.

It will be unnecessary to recapitulate all the advantages of this invention. Mention has been .made, for example, of the desirability of having several coils on a single core. It is thus possible to have the same number of arbitrary constants in the circuit, though using fewer elements and occupying less space. The counting of several coils on a single core as one element has already been referred to.

Again, if a given network is to be reduced to an equivalent network more suitable for practical design,as, for example, having less elements,- it is often not the best way to apply the principle of linear transformation directly to the circuit. In all cases where the given circuit has not the least number of independent meshes, we replace, first, the network by any equivalent network which has the least number of elements. For instance, if we have a bridge-type, four terminal network,as the circuit, Fig. '7, input terminals 1, 1 and output terminals 50, 52, with the resistance R. removed,and arbitrary 21 and 22 without resistances, then a first equivalent network without superfluous meshes may be obtained in the following way: We use the relations Z1+Z2 zz 2 im and As the network is symmetrical,

The procedure from this point on for obtaining a first network without superfluous'meshes is the same as before described.

As a further example, if we have several bridge- 'not necessarily constant, we can reduce this circuit to a single, bridge-type circuit according to the procedure on page 279 of my paper entitled 1,958,742 7 Vierpole in Elektrische Nachrichtentechnik, If the meshes II, III and IV are oriented as 1929. in Fig. 14, and assuming arbitrarily, but pre- The invention will now be further explained missibly, that mesh II is not capacitatively conin connection with Figs. 11 to 20. nected with the meshes III and IV, so as to have Let it be assumed that it is desired to obtain D D D 80 a more economical, balanced network equivalent 31 to the symmetrical, four-terminal, reactive netth at s st) and st f t final n w k, work illustr ted in Fig, 11, having f r br hes, with four meshes, will have the following values: as in Fig. 8,'with the respective impedance Z1 and L L L L Z2, as indicated in the said Fig. 11. The im- 0 s pedances Z1 and Z2 are each a two-terminal por- (L82): L 1,; LH-L L tion, as represented in Figs. 12 and 13, respec- L 0 L L +L tively. The two-terminal portion Z1, has series D D 0 0 capacitances 80, 82 and 84 and shunt inductances 2: 2 0 o 81, 83 and 85. The two-terminal portion Z2 has (DH): 0 D3 0 a series capacitance 86 and a shunt inductance 87. 0 0 0 D Let the reclprocal values of the capacltances To determine D12, L12, L13, L14, the coeificients of 82, 84 and 86 be represented by d1, d2, d3 and d4, the various, powersvof in an may be respectively. Let the values of the inductances pared be represented by Z1, Z2, Z3 and Z4, respectively.

Following out the procedure outlined above, the ma 2+ a) 3 a first step is to construct a twd-terminal circuit for Lax the secondary open-circuit impedance, that is, the +1,1 2 impedance obtained across the terminals 2, 2' of {+[(I1+I2)d3+(Iz+I3)dg]X+(12113} Fig. 11, the terminals 1, 1' being disregarded, as Where 7 though they were open, with the modification, however, that the secondary open-circuit im- (11+ 0 G1 24 2 4-1 4 1) pedance will be taken now instead of the primary 41 ,1 Open clrcult Impedance: Since, from the network of Fig. 11,

fi it l. 8 .Z1Z2 1122 2 fi T Instead of resolving it follows that 11 E lz=g( o 1 z a) in partial fractions, which would result in a spewhere cial case of the circuit illustrated in Fig. 1, it is preferred here to develop in a Stieltjes continued =1,1 1 (1 1 -{-1,1,-1-1,1' )I fraction as explained for example, in my paper e =I l (d +d +d )+I I (d +d )+1 I3d1 entitled, Verwirklichung von Wechselstromi[( 1+ z) 3+( 2+ a) 2] 4+( z a+ a 1+ 1 0 4} widerstanden vorgeschriebener Frequenzabhang- 2=. 1( 1+ 2) a+ 2( 2+ s) 1+ a r 2 4 z a+ igkeit, in Archiv fiir Electrotechnik, Heft 1 4, [(11+12)d3+(12+13)d21d4i Band XVII, 1920. The two-terminal network eszdlhdaidzdan shown by heavy lines in Fig. 14, and having the on the other hand, according to the definition of two terminals 2, 2, is thus obtained. The ree I maining circuit elements of the final network are an glven m the gen m1 treatment above next calculated, as also described above. The L Do L L first four-terminal network thus obtained, as an wan: g; +1 g: equivalent to the four-terminal network illus- 0 Lax (133+ D4 trated in Fig. 11, will be that illustrated in Fig. 14.

To obtain the said heavy-line, two-terminal L (L L L L LgLg) L13L2(L3 L4) network of Fig. 14, bearing in mind that Z1 and L L L ]x Z2 are as illustrated in Figs. 12 and 13, the follow- 12( 2 a+ 3 4+ 4 1)+ 12[( 2+ a) 4+ K K 1 a 2121 12 s 4= a= d2 a( 1-d 135 L 1 z+ 2 4+ 1 1)[ 1 4( 2+ s)+ 2 a( 1+ 4)] 21 ,1 from which it follows that 2 Da l+ 4) 2 D12 D21 d4 d1 21 1 2 D (l112+12I4+l411)2d3 A similar comparison of the coefiicients of x 32?,

4- Y 1 2 :2 1n Mam gives sim larly To simplify the notation, the following are introduced also: L12=L27= 2 2 1+ 4) LBLTLM: (I1:I4)= L2 o 1 2 3 4 L=L=Q 11tuz+13 15130191) a To calculate the remaining unknowns, L11 and D11, a similar comparison is made of the coefficients of the different powers of a: in was, taking 1122 from the original network, In the particular case of Fig. 11, the original network happens to be symmetrical, so that 1y. The inductance 81 of Fig. 12 becomes the inductance 102 of Fig. 15, with the same value 11.

In this special case, however, no change occurs in the connections of Fig. 13 which may, however, be represented as in Fig. 16, the condensers 86 being there illustrated at 103 and the inductance 87 at 104.

The network of Fig. 14 thus becomes reduced to the special case illustrated in Fig. 17.

an=az2 s a, check upon the work, and to give an illusfrom which it follows that tration of the method of matrix multiplication 31 2+ 0 3 3 2 z'if s) s a u a (3 O 4 0 3 3 4) 4 A comparison of the coefficient of x on both sides yields Du=D2,

and a similar comparison of the coefficients'of x yields, finally,

The resulting matrices (Lst) and (Dst) of the elec-. trict constants of the different meshes are now Inspection of Fig. 14 demonstrates that the network illustrated therein fulfills the conditions required by these electric constants, with tight couplings. Thus, the mutual inductance between the meshes I and III is and the self-inductance of mesh I is The coils 94 and 95 are tightly coupled, with a turn-ratio The turn-ratio of the tightly coupled coils 97 and 98 is I Z4 2 L4. The turn-ratio may be regarded as positive when the inductances of a pair of coupled coils are additive when connected in series in the same direction.

' In thespecial case when the connections of Fig. 12 reduce to those of Fig.15. The condensers 80 and '82 of Fig. .12 become the condensers 100 and 101 of Fig. 15, with the same reciprocal values d1, d2 respectivereferred to in the general treatment above, the

equivalence of the networks illustrated in Figs. 11 and 17, for the special case where.

g; g o

d|+dg+d4 d d all 2 2 2 ot) drgdl g- 0.

Equivalent circuits, with new constants L'st and Us, may be obtained by means of the relations I I I n e t at a t st a t by the transformation which is obtained from the general transformation discussed in an earlier part of this specification, letting It results that the (Lst) and (D'n) may be calculated by the following matrix multiplication:

(UH): g g 4 g ofthe mesh containing it, and if the turn-ratio is g 9 7 L32 1, 5 7 positive. Each change in there orientations, or in L11 201L13+ aa L12 1" 1 23 1" BL13+ l a3 7 :13 Y as iz+l 1a+ 2a+ 5 aa 42+ n-P5 33 'Y za-l'fir aa 7 l3+ 7 33 7 43+ I3'YL33 7 43 An analogous matrix may be obtained for (Dsz). the sign of the turn-ratio, involves the introduc- 8 a, ,8, 7 may be determined by the assumptions tion of one minus sign for Lu. For instance, if both meshes have the opposite orientation to that iz of the corresponding coils contained therein, and 2723:0 if the turn-ratio is negative, three negative signs are involved, so that, therefore, L12 is negative in which conditions are satisfied 1n the network this case It follows that The numbers 4, 2.5 etc. on the coils 128, 130, etc.

are proportional to the turn number. a: F Therefore, the turn-ratio between the coils 128 Putting in these values and the above values of a It Onows that Fig. 18 for the (Let) and (Dat) it results that: 2

g 04x6 =0.0390625 henry.

v The self -inductivit of mesh III is a-r 11+! I1+I4 y 2 2 Z 2 Lf 3=4 0.0390625+ 0.05625 2l ALL 0.78125 henry. d d d d 1 As an example of mutual inductance, let L'n be 05 0 calculated. According to the above explanations,

l oh-d; d d-d LI: '4 0-05625 E= 0.375- (D at) 2 2 0 I '5 +10%: Though the networks of Figs. 19 and 20 are 0 0 T equivalent, and thus attain, or approximate to,

These circuit elements are found in the network of Fig. 17, which is thus equivalent to that of Fig. 18 which, in turn, is equivalent to that of Fig. 11. The second calculation has independently thus yielded the same results as the first calculation.

The network of Fig. 19 is a special case of the network of Fig. 11, for the following special values:

d =I =0 d3 =0.1l4F (14 0.4111 I =1H 1 :0.25H.

This network represents a known form of phasecornpensating network.

The network of Fig. 20 is an equivalent network of the same type as that of Fig. 19, with mutual inductanoes all in the form of tight couplings. For example, the coils 128, 130 and 131 are mutually inductively related in this manner.

The numerical values annexed to the various parts in Fig. 20 make it easy to read from the diagram that the circuit elements are determined by the matrices:

The mutual inductance L12 between meshes I and II may be assumed positive if the orientation of the coils has the same sense as the orientation the same, predetermined, frequency characteristics, no two of the three mutually inductively related coils 128, 134 and 136'of the network of Fig. 20 are directly connected together galvanically. No such arrangement is disclosed in Fig. 19. The employment of this mutual inductance, as illustrated, for example, in Fig. 20, thus enables one to attain equally good, or better, results, but with greater economy, involving the employment of a fewer number of circuit elements. This is not true of the network of Fig. 19. One glance sufiices to show how much more economical is the network of Fig. 20.

The network of Fig. 20 is not quite a special 1 case of that of Fig. 17, as it is somewhat changed, so as to be balanced.

It will be understood that the invention is not restricted to the exact embodiments herein illustrated and described, but that modifications may be made by persons skilled in the art, and all such are considered to fall within the scope of this invention. a

What I claim is: v

1. An artificial network attaining or approximating predetermined frequency characteristics and having one or more passive network sections, one of the sections having three or more mutually I inductively related coils no two of which are directly galvanically connected together. 4

2. Anartificial network attaining or approximating predetermined frequency characteristics and having one or more passive network sections, one of the sections having three or more mutually inductively related coils tightly coupled together no two of which are directly galvanically condisposed portions being substantially alike, the product of the unlike portions being substantially equal to the square of a resistance across two of the vertices.

4. An artificial network having one or more passive network sections and provided with two terminals only, one of the sections having three or more mutually inductively related coils no two of which are directly galvanically connected together, and means connecting the coils with the terminals.

5. An artificial network having. one or more passive network sections including a two-ter-' minal network portion provided with three or more mutually inductively related coils no two of which are directly galvanically connected together, and means connecting the coils with the terminals.

6. An artificial network having one or more passive network sections and providedwith two terminals only, one of the sections having three or more mutually inductively related coils tightly coupled together, no two of which are directly galvanically connected together, and means connecting the coils with the terminals.

7. An artificial network having one or more passive network sections including a two-terminal network portion provided with three or more mutually inductively related coils tightly coupled together, no two of which are directly galvanicallyconnected together, and means connecting the coils with the terminals.

8. An artificial network having four terminals only and provided with one or more passive network sections, one of the sections having three or more mutually inductively related coils no two of which are directly galvanically connected together, and means connecting the coils with the terminals.

9. An artificial network having one or more passive network sections including a four-terminal network portion, provided with three or more mutually inductively related coils no two of which are directly galvanically connected together, and means connecting the coils with the terminals.

10. An artificial network having four terminals only and provided with one or more passive, network sections, one of the sections having three or more mutually inductively related coils tightly coupledv together, no two of which are directly galvanically connected together; and means con necting the coils with the terminals.

11. An artificial network having one or more passive, network sections including a four-terminal network portion provided with three or more mutually inductively related coils tightly coupled together, no twoof which are directly galvanically connected together, and means connecting the coils with the terminals.

12. Anartificial network attaining or approximating predetermined frequency characteristics and having one or more passive network sections, one of the sections having three or more mutually inductively related coils no two of which are directly galvanically connected together, the network being balanced with respect to the ground potential.

13. An artificial network attaining or approximating predetermined frequency characteristics and having one\ or more passive network setherein, the characteristic matrix of the impedtions, one of the sections having three or more mutually inductively related coils tightly coupled together no two of which are directly galvanically connected together, the network being balanced with respect to the ground potential.

14. An artificial network having four terminals only and provided with one or more passive network sections, one of the sections having three or more mutually inductively related coils no two of which are directly galvanically connected together, and means connecting the coils with the terminals, the network being balanced with respect to the ground potential.

15. An artificial network having one or more passive network sections including a four-terminal network portion provided with three or more mutually inductively related coils no two of which are directly galvanically connected together, and means connecting the coils with the terminals, the network being balanced with respect to the ground potential.

16. An artificial network having four terminals only and provided with one or more passive network sections, one of the sections having three or more mutually inductively related coils tightly coupled together, no two of which are directly galvanically connected together, and means connecting the coils with the terminals, the network being balanced with respect to the ground potential.

17. An artificial network having one or more passive network sections including a four-terminal network portion provided with three or more mutually inductively related coils tightly coupled together, and means connecting the coils with the terminals, the network being balanced with respect to the ground potential.

18. An artificial network having two network portions each provided with mutual inductances ance forming one of said network portions being the reciprocal of the characteristic matrix of the impedances forming said other network portion, multiplied by a positive, constant factor.

19. An artificial network having two network portions each provided with mutual inductances thereinand each having a number of pairs of terminals, the characteristic matrix of the impedances forming one of said network portions being the reciprocal of the characteristic matrix 1 of the impedances forming said othernetwork portion, multiplied by a positive, constant factor.

20. An artificial network havingtwo network portions each provided with mutual inductances therein tightly coupled together, the characteristic matrix of the impedances forming one of said network portions being the reciprocal of the characteristic matrix of the impedances forming said other network portion, multiplied by a positive, constant factor.

21. An artificial network having two network portions each provided with mutual inductances therein tightly coupled together and each having a number of pairs of terminals, the characteristic matrix of the impedances forming one of said 14 network portions being the reciprocal. of the characteristic matrix of the impedances forming said other network portion, multiplied by a positive, constant factor.

' WILHELM CAUER. 

